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Theorem zfcndun 9638
Description: Axiom of Union ax-un 7095, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
Assertion
Ref Expression
zfcndun 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem zfcndun
StepHypRef Expression
1 axunnd 9619 . 2 𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦)
2 elequ2 2158 . . . . . . 7 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
3 elequ1 2151 . . . . . . 7 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
42, 3anbi12d 608 . . . . . 6 (𝑤 = 𝑦 → ((𝑧𝑤𝑤𝑥) ↔ (𝑧𝑦𝑦𝑥)))
54cbvexv 2434 . . . . 5 (∃𝑤(𝑧𝑤𝑤𝑥) ↔ ∃𝑦(𝑧𝑦𝑦𝑥))
65imbi1i 338 . . . 4 ((∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ (∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
76albii 1894 . . 3 (∀𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ ∀𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
87exbii 1923 . 2 (∃𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦) ↔ ∃𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
91, 8mpbir 221 1 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1628   = wceq 1630  wex 1851  wcel 2144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095  ax-reg 8652
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-eprel 5162  df-fr 5208
This theorem is referenced by: (None)
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